4 Section 3.1 ObjectivesIdentify the sample space of a probability experimentIdentify simple eventsUse the Fundamental Counting PrincipleDistinguish among classical probability, empirical probability, and subjective probabilityDetermine the probability of the complement of an eventUse a tree diagram and the Fundamental Counting Principle to find probabilitiesLarson/Farber 4th ed

5 Probability ExperimentsAn action, or trial, through which specific results (counts, measurements, or responses) are obtained.OutcomeThe result of a single trial in a probability experiment.Sample SpaceThe set of all possible outcomes of a probability experiment.EventConsists of one or more outcomes and is a subset of the sample space.Larson/Farber 4th ed

7 Example: Identifying the Sample SpaceA probability experiment consists of tossing a coin and then rolling a six-sided die. Describe the sample space.Solution:There are two possible outcomes when tossing a coin: a head (H) or a tail (T). For each of these, there are six possible outcomes when rolling a die: 1, 2, 3, 4, 5, or 6. One way to list outcomes for actions occurring in a sequence is to use a tree diagram.Larson/Farber 4th ed

9 Simple Events Simple event An event that consists of a single outcome.e.g. “Tossing heads and rolling a 3” {H3}An event that consists of more than one outcome is not a simple event.e.g. “Tossing heads and rolling an even number” {H2, H4, H6}Larson/Farber 4th ed

10 Example: Identifying Simple EventsDetermine whether the event is simple or not.You roll a six-sided die. Event B is rolling at least a 4.Solution:Not simple (event B has three outcomes: rolling a 4, a 5, or a 6)Larson/Farber 4th ed

11 Fundamental Counting PrincipleIf one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence is m*n.Can be extended for any number of events occurring in sequence.Larson/Farber 4th ed

18 Example: Finding Empirical ProbabilitiesA company is conducting an online survey of randomly selected individuals to determine if traffic congestion is a problem in their community. So far, 320 people have responded to the survey. What is the probability that the next person that responds to the survey says that traffic congestion is a serious problem in their community?ResponseNumber of times, fSerious problem123Moderate problem115Not a problem82Σf = 320Larson/Farber 4th ed

20 Law of Large Numbers Law of Large NumbersAs an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event.Larson/Farber 4th ed

21 Types of Probability Subjective ProbabilityIntuition, educated guesses, and estimates.e.g. A doctor may feel a patient has a 90% chance of a full recovery.Larson/Farber 4th ed

22 Example: Classifying Types of ProbabilityClassify the statement as an example of classical, empirical, or subjective probability.The probability that you will be married by age 30 is 0.50.Solution:Subjective probability (most likely an educated guess)Larson/Farber 4th ed

23 Example: Classifying Types of ProbabilityClassify the statement as an example of classical, empirical, or subjective probability.The probability that a voter chosen at random will vote Republican is 0.45.Solution:Empirical probability (most likely based on a survey)Larson/Farber 4th ed

24 Example: Classifying Types of ProbabilityClassify the statement as an example of classical, empirical, or subjective probability.The probability of winning a 1000-ticket raffle with one ticket isSolution:Classical probability (equally likely outcomes)Larson/Farber 4th ed

27 Example: Probability of the Complement of an EventYou survey a sample of 1000 employees at a company and record the age of each. Find the probability of randomly choosing an employee who is not between 25 and 34 years old.Employee agesFrequency, f15 to 245425 to 3436635 to 4423345 to 5418055 to 6412565 and over42Σf = 1000Larson/Farber 4th ed

28 Solution: Probability of the Complement of an EventUse empirical probability to find P(age 25 to 34)Employee agesFrequency, f15 to 245425 to 3436635 to 4423345 to 5418055 to 6412565 and over42Σf = 1000Use the complement ruleLarson/Farber 4th ed

29 Example: Probability Using a Tree DiagramA probability experiment consists of tossing a coin and spinning the spinner shown. The spinner is equally likely to land on each number. Use a tree diagram to find the probability of tossing a tail and spinning an odd number.Larson/Farber 4th ed

31 Example: Probability Using the Fundamental Counting PrincipleYour college identification number consists of 8 digits. Each digit can be 0 through 9 and each digit can be repeated. What is the probability of getting your college identification number when randomly generating eight digits?Larson/Farber 4th ed

33 Section 3.1 SummaryIdentified the sample space of a probability experimentIdentified simple eventsUsed the Fundamental Counting PrincipleDistinguished among classical probability, empirical probability, and subjective probabilityDetermined the probability of the complement of an eventUsed a tree diagram and the Fundamental Counting Principle to find probabilitiesLarson/Farber 4th ed

36 Conditional ProbabilityThe probability of an event occurring, given that another event has already occurredDenoted P(B | A) (read “probability of B, given A”)Larson/Farber 4th ed

37 Example: Finding Conditional ProbabilitiesTwo cards are selected in sequence from a standard deck. Find the probability that the second card is a queen, given that the first card is a king. (Assume that the king is not replaced.)Solution:Because the first card is a king and is not replaced, the remaining deck has 51 cards, 4 of which are queens.Larson/Farber 4th ed

38 Example: Finding Conditional ProbabilitiesThe table shows the results of a study in which researchers examined a child’s IQ and the presence of a specific gene in the child. Find the probability that a child has a high IQ, given that the child has the gene.Gene PresentGene not presentTotalHigh IQ331952Normal IQ3911507230102Larson/Farber 4th ed

39 Solution: Finding Conditional ProbabilitiesThere are 72 children who have the gene. So, the sample space consists of these 72 children.Gene PresentGene not presentTotalHigh IQ331952Normal IQ3911507230102Of these, 33 have a high IQ.Larson/Farber 4th ed

40 Independent and Dependent EventsIndependent eventsThe occurrence of one of the events does not affect the probability of the occurrence of the other eventP(B | A) = P(B) or P(A | B) = P(A)Events that are not independent are dependentLarson/Farber 4th ed

41 Example: Independent and Dependent EventsDecide whether the events are independent or dependent.Selecting a king from a standard deck (A), not replacing it, and then selecting a queen from the deck (B).Solution:Dependent (the occurrence of A changes the probability of the occurrence of B)Larson/Farber 4th ed

42 Example: Independent and Dependent EventsDecide whether the events are independent or dependent.Tossing a coin and getting a head (A), and then rolling a six-sided die and obtaining a 6 (B).Solution:Independent (the occurrence of A does not change the probability of the occurrence of B)Larson/Farber 4th ed

43 The Multiplication RuleMultiplication rule for the probability of A and BThe probability that two events A and B will occur in sequence isP(A and B) = P(A) ∙ P(B | A)For independent events the rule can be simplified toP(A and B) = P(A) ∙ P(B)Can be extended for any number of independent eventsLarson/Farber 4th ed

44 Example: Using the Multiplication RuleTwo cards are selected, without replacing the first card, from a standard deck. Find the probability of selecting a king and then selecting a queen.Solution:Because the first card is not replaced, the events are dependent.Larson/Farber 4th ed

45 Example: Using the Multiplication RuleA coin is tossed and a die is rolled. Find the probability of getting a head and then rolling a 6.Solution:The outcome of the coin does not affect the probability of rolling a 6 on the die. These two events are independent.Larson/Farber 4th ed

46 Example: Using the Multiplication RuleThe probability that a particular knee surgery is successful is Find the probability that three knee surgeries are successful.Solution:The probability that each knee surgery is successful is The chance for success for one surgery is independent of the chances for the other surgeries.P(3 surgeries are successful) = (0.85)(0.85)(0.85)≈ 0.614Larson/Farber 4th ed

47 Example: Using the Multiplication RuleFind the probability that none of the three knee surgeries is successful.Solution:Because the probability of success for one surgery is The probability of failure for one surgery is 1 – 0.85 = 0.15P(none of the 3 surgeries is successful) = (0.15)(0.15)(0.15)≈ 0.003Larson/Farber 4th ed

48 Example: Using the Multiplication RuleFind the probability that at least one of the three knee surgeries is successful.Solution:“At least one” means one or more. The complement to the event “at least one successful” is the event “none are successful.” Using the complement ruleP(at least 1 is successful) = 1 – P(none are successful)≈ 1 – 0.003= 0.997Larson/Farber 4th ed

49 Example: Using the Multiplication Rule to Find ProbabilitiesMore than 15,000 U.S. medical school seniors applied to residency programs in Of those, 93% were matched to a residency position. Seventy-four percent of the seniors matched to a residency position were matched to one of their top two choices. Medical students electronically rank the residency programs in their order of preference and program directors across the United States do the same. The term “match” refers to the process where a student’s preference list and a program director’s preference list overlap, resulting in the placement of the student for a residency position. (Source: National Resident Matching Program)(continued)Larson/Farber 4th ed

50 Example: Using the Multiplication Rule to Find ProbabilitiesFind the probability that a randomly selected senior was matched a residency position and it was one of the senior’s top two choices.Solution:A = {matched to residency position}B = {matched to one of two top choices}P(A) = and P(B | A) = 0.74P(A and B) =P(A)∙P(B | A) = (0.93)(0.74) ≈ 0.688dependent eventsLarson/Farber 4th ed

51 Example: Using the Multiplication Rule to Find ProbabilitiesFind the probability that a randomly selected senior that was matched to a residency position did not get matched with one of the senior’s top two choices.Solution:Use the complement:P(B′ | A) = 1 – P(B | A)= 1 – = 0.26Larson/Farber 4th ed

54 Section 3.3 Objectives Determine if two events are mutually exclusiveUse the Addition Rule to find the probability of two eventsLarson/Farber 4th ed

55 Mutually Exclusive EventsTwo events A and B cannot occur at the same timeABABA and B are mutuallyexclusiveA and B are not mutually exclusiveLarson/Farber 4th ed

56 Example: Mutually Exclusive EventsDecide if the events are mutually exclusive. Event A: Roll a 3 on a die. Event B: Roll a 4 on a die.Solution:Mutually exclusive (The first event has one outcome, a 3. The second event also has one outcome, a 4. These outcomes cannot occur at the same time.)Larson/Farber 4th ed

58 The Addition Rule Addition rule for the probability of A or BThe probability that events A or B will occur isP(A or B) = P(A) + P(B) – P(A and B)For mutually exclusive events A and B, the rule can be simplified toP(A or B) = P(A) + P(B)Can be extended to any number of mutually exclusive eventsLarson/Farber 4th ed

59 Example: Using the Addition RuleYou select a card from a standard deck. Find the probability that the card is a 4 or an ace.Solution:The events are mutually exclusive (if the card is a 4, it cannot be an ace)4♣4♥4♦4♠A♣A♥A♦A♠44 other cardsDeck of 52 CardsLarson/Farber 4th ed

60 Example: Using the Addition RuleYou roll a die. Find the probability of rolling a number less than 3 or rolling an odd number.Solution:The events are not mutually exclusive (1 is an outcome of both events)Odd531246Less than threeRoll a DieLarson/Farber 4th ed

61 Solution: Using the Addition RuleOdd531246Less than threeRoll a DieLarson/Farber 4th ed

62 Example: Using the Addition RuleThe frequency distribution shows the volume of sales (in dollars) and the number of months a sales representative reached each sales level during the past three years. If this sales pattern continues, what is the probability that the sales representative will sell between $75,000 and $124,999 next month?Sales volume ($)Months0–24,999325,000–49,999550,000–74,999675,000–99,9997100,000–124,9999125,000–149,9992150,000–174,999175,000–199,9991Larson/Farber 4th ed

64 Example: Using the Addition RuleA blood bank catalogs the types of blood given by donors during the last five days. A donor is selected at random. Find the probability the donor has type O or type A blood.Type OType AType BType ABTotalRh-Positive1561393712344Rh-Negative282584651841644516409Larson/Farber 4th ed

66 Example: Using the Addition RuleFind the probability the donor has type B or is Rh-negative.Type OType AType BType ABTotalRh-Positive1561393712344Rh-Negative282584651841644516409Solution:The events are not mutually exclusive (a donor can have type B blood and be Rh-negative)Larson/Farber 4th ed

70 Section 3.4 ObjectivesDetermine the number of ways a group of objects can be arranged in orderDetermine the number of ways to choose several objects from a group without regard to orderUse the counting principles to find probabilitiesLarson/Farber 4th ed

72 Example: Permutation of n ObjectsThe objective of a 9 x 9 Sudoku number puzzle is to fill the grid so that each row, each column, and each 3 x 3 grid contain the digits 1 to 9. How many different ways can the first row of a blank 9 x 9 Sudoku grid be filled?Solution:The number of permutations is 9!= 9∙8∙7∙6∙5∙4∙3∙2∙1 = 362,880 waysLarson/Farber 4th ed

73 Permutations Permutation of n objects taken r at a timeThe number of different permutations of n distinct objects taken r at a time■where r ≤ nLarson/Farber 4th ed

74 Example: Finding nPrFind the number of ways of forming three-digit codes in which no digit is repeated.Solution:You need to select 3 digits from a group of 10n = 10, r = 3Larson/Farber 4th ed

75 Example: Finding nPrForty-three race cars started the 2007 Daytona 500. How many ways can the cars finish first, second, and third?Solution:You need to select 3 cars from a group of 43n = 43, r = 3Larson/Farber 4th ed

76 Distinguishable PermutationsThe number of distinguishable permutations of n objects where n1 are of one type, n2 are of another type, and so on■where n1 + n2 + n3 +∙∙∙+ nk = nLarson/Farber 4th ed

78 Combinations Combination of n objects taken r at a timeA selection of r objects from a group of n objects without regard to order■Larson/Farber 4th ed

79 Example: CombinationsA state’s department of transportation plans to develop a new section of interstate highway and receives 16 bids for the project. The state plans to hire four of the bidding companies. How many different combinations of four companies can be selected from the 16 bidding companies?Solution:You need to select 4 companies from a group of 16n = 16, r = 4Order is not importantLarson/Farber 4th ed

81 Example: Finding ProbabilitiesA student advisory board consists of 17 members. Three members serve as the board’s chair, secretary, and webmaster. Each member is equally likely to serve any of the positions. What is the probability of selecting at random the three members that hold each position?Larson/Farber 4th ed

82 Solution: Finding ProbabilitiesThere is only one favorable outcomeThere areways the three positions can be filledLarson/Farber 4th ed

83 Example: Finding ProbabilitiesYou have 11 letters consisting of one M, four Is, four Ss, and two Ps. If the letters are randomly arranged in order, what is the probability that the arrangement spells the word Mississippi?Larson/Farber 4th ed

84 Solution: Finding ProbabilitiesThere is only one favorable outcomeThere aredistinguishable permutations of the given letters11 letters with 1,4,4, and 2 like lettersLarson/Farber 4th ed

85 Example: Finding ProbabilitiesA food manufacturer is analyzing a sample of 400 corn kernels for the presence of a toxin. In this sample, three kernels have dangerously high levels of the toxin. If four kernels are randomly selected from the sample, what is the probability that exactly one kernel contains a dangerously high level of the toxin?Larson/Farber 4th ed

86 Solution: Finding ProbabilitiesThe possible number of ways of choosing one toxic kernel out of three toxic kernels is3C1 = 3The possible number of ways of choosing three nontoxic kernels from 397 nontoxic kernels is397C3 = 10,349,790Using the Multiplication Rule, the number of ways of choosing one toxic kernel and three nontoxic kernels is3C1 ∙ 397C3 = 3 ∙ 10,349,790 3 = 31,049,370Larson/Farber 4th ed

88 Section 3.4 SummaryDetermined the number of ways a group of objects can be arranged in orderDetermined the number of ways to choose several objects from a group without regard to orderUsed the counting principles to find probabilitiesLarson/Farber 4th ed

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